Semiclassical Limit of the Nonlinear Schr$#x00F6;dinger-Poisson Equation with Subcritical Initial Data

Abstract

We study the semi-classical limit of the nonlinear Schr$#x00F6;dinger-Poisson (NLSP) equation for initial data of the WKB type. The semi-classical limit in this case is realized in terms of a density-velocity pair governed by the Euler-Poisson equations. Recently we have shown that the isotropic Euler-Poisson equations admit a critical threshold phenomena, where initial data in the sub-critical regime give rise to globally smooth solutions. Consequently, we justify the semi-classical limit for sub-critical NLSP initial data and confirm the validity of the WKB method.